关于特定型极限limx→+∞1/xk∫x0t^(k-1)︱cos t︱dt的分析与计算

On Analysis and Calculation of a Specific Limit limx→+∞1/xk∫x0t^(k-1)︱cos t︱dt

通过引入伯努利数和自然数等幂和公式,给出相应的奇自然数等幂和公式.利用夹逼定理将该特定型函数极限转化成商式数列的极限问题,并由第二数学归纳法推出分子表达式的不定积分表示,进而应用余弦函数的周期性和奇自然数等幂和公式推得分子表达式的通项公式,再由夹逼定理获得该特定型极限的一般结果,从而回答了所提出猜想是正确的.

The formula on the sum of equal power of odd natural numbers is given by Bernoulli numbers and the formula on the sum of equal power of natural numbers. The specific function limit is transformed into that of a sequence of quotient by approximation theorem, and the representation with respect to indefinite integral of the numerator is obtained by the second mathematical induction. The general expression of numerator is derived from the periodicity of cosine function and the formula on the sum of equal power of odd natural numbers, and using approximation theorem , this paper get the general result of the specific limit, which shows that the guess is correct.